Test to see if ΔFEG is a right triangle.

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1 1. Copy the figure shown, and draw the common tangents. If no common tangent exists, state no common tangent. Every tangent drawn to the small circle will intersect the larger circle in two points. Every tangent drawn to the large circle will not intersect the small circle at any point. Since a tangent must intersect the circle at exactly one point, no common tangent exists for these two circles. no common tangent Determine whether is tangent to. Justify your answer. 3. Test to see if ΔEFG is a right triangle. Yes; ΔEFG is a right triangle, so circle E. yes; 1521 = 1521 is tangent to Find x. Assume that segments that appear to be tangent are tangent. 2. Test to see if ΔFEG is a right triangle. 4. By Theorem 10.10,. So, is a right triangle. No; ΔFEG is not a right triangle, so tangent to circle E. is not No; 20 esolutions Manual - Powered by Cognero Page 1

2 5. 7. LANDSCAPE ARCHITECT A landscape architect is paving the two walking paths that are tangent to two approximately circular ponds as shown. The lengths given are in feet. Find the values of x and y. By Theorem 10.10,. So, is a right triangle. 16 If two segments from the same exterior point are tangent to a circle, then they are congruent. To find the value of x, use the lengths of the two sidewalk segments that are tangent to the smaller pond. 6. To find the value of y, use the total lengths of the two sidewalk segments that are tangent to the larger pond and substitute 250 for the value of x. If two segments from the same exterior point are tangent to a circle, then they are congruent. Therefore, x = 250 and y = 275. x = 250; y = esolutions Manual - Powered by Cognero Page 2

3 8. CCSS SENSE-MAKING Triangle JKL is circumscribed about. Copy each figure and draw the common tangents. If no common tangent exists, state no common tangent. a. Find x. b. Find the perimeter of. a. If two segments from the same exterior point are tangent to a circle, then they are congruent. 9. Three common tangents can be drawn. b. Since two tangent segments from the same exterior point are congruent, JM = JO = 12, KN = KM = 7, and LO = LN = or 7. The sides of the triangle will have lengths of JK = or 19, KL = or 14, and JL = or 19. To find the perimeter of a triangle, add the lengths of its sides. Therefore, the perimeter of triangle JKL is 52 units. a. 4 b. 52 units 10. Every tangent drawn to the small circle will intersect the larger circle in two points. Every tangent drawn to the large circle will not intersect the small circle at any point. Since a tangent must intersect the circle at exactly one point, no common tangent exists for these two circles. no common tangent esolutions Manual - Powered by Cognero Page 3

4 11. Four common tangents can be drawn to these two circles. 12. Two common tangents can be drawn to these two circles. Determine whether each is tangent to the given circle. Justify your answer. 13. Yes; Yes; 625 = 625 esolutions Manual - Powered by Cognero Page 4

5 Find x. Assume that segments that appear to be tangent are tangent. Round to the nearest tenth if necessary. 14. Yes; 17. Yes; 100 = 100 By Theorem 10.10,. So, is a right triangle. Use the Pythagorean Theorem. Substitute. 15. No; 26 No; yes; By Theorem 10.10,. So, is a right triangle. Use the Pythagorean Theorem. Substitute. Yes; 80 = esolutions Manual - Powered by Cognero Page 5

6 19. By Theorem 10.10,. So, is a right triangle. Use the Pythagorean Theorem. 21. If two segments from the same exterior point are tangent to a circle, then they are congruent. Substitute If two segments from the same exterior point are tangent to a circle, then they are congruent. 20. By Theorem 10.10,. So, is a right triangle. Use the Pythagorean Theorem. 1 Substitute esolutions Manual - Powered by Cognero Page 6

7 23. ARBORS In the arbor shown, and are tangents to, and points D, E, and C are collinear. The radius of the circle is 26 inches and EC = 20 inches. Find each measure to the nearest hundredth. CCSS SENSE-MAKING Find the value of x. Then find the perimeter. a. AC b. BC a. Draw triangle DAC. 24. Find the missing measures. Since is circumscribed about, and are tangent to, as are,,, and. Therefore,,, and. So, QU = QT. 2x = 14 x = 7 TS = VS = 17 RU = RV = = 10 To find the perimeter of a triangle, add the lengths of its sides. By Theorem 10.10,. So, is a right 7; 82 in. triangle. Since the radius of is 26, AD = 26 inches. By the Segment Addition Postulate, DC = DE + EC. So, DC = or 46 inches. Use the Pythagorean Theorem to find AC. Therefore, the measure of AC is about inches. b. If two segments from the same exterior point are tangent to a circle, then they are congruent. and are both tangent to from point C. So,. a in. b in. esolutions Manual - Powered by Cognero Page 7

8 Find x to the nearest hundredth. Assume that segments that appear to be tangent are tangent. 25. Since quadrilateral ABCD is circumscribed about, and are tangent to, as are,,,,, and. Therefore,, So, AM = AL.,,and. AM = 5 MB = 13 5 = 8 x = BN = MB = 8 LD = DP = 6 PC = NC = 7 To find the perimeter of a triangle, add the lengths of its sides. 26. If two segments from the same exterior point are tangent to a circle, then they are congruent. Here, TS = TR and TR = TQ. By the Transitive Property, TS = TQ. 9 8; 52 cm 27. QS = = 9 By Theorem 10.10,. So, is a right triangle. Use the Pythagorean Theorem. Substitute. If two segments from the same exterior point are tangent to a circle, then they are congruent. So, esolutions Manual - Powered by Cognero Page 8

9 Write the specified type of proof. 28. two-column proof of Theorem Given: is tangent to at C. is tangent to at B. Prove: about. Prove: AB + CD = AD + BC Proof: Statements (Reasons) 1. is tangent to at C; is tangent to at B. (Given) 2. Draw,, and. (Through any two points, there is one line.) 3., (Line tangent to a circle is to the radius at the pt. of tangency.) 4. and are right angles. (Def. of lines) 5. (All radii of a circle are.) 6. (Reflexive Prop.) 7. (HL) 8. (CPCTC) Proof: Statements (Reasons) 1. is tangent to at C; is tangent to at B. (Given) 2. Draw,, and. (Through any two points, there is one line.) 3., (Line tangent to a circle is to the radius at the pt. of tangency.) 4. and are right angles. (Def. of lines) 5. (All radii of a circle are.) 6. (Reflexive Prop.) 7. (HL) 8. (CPCTC) 29. two-column proof Given: Quadrilateral ABCD is circumscribed Statements (Reasons) 1. Quadrilateral ABCD is circumscribed about. (Given) 2. Sides,,, and are tangent to at points H, G, F, and E, respectively. (Def. of circumscribed) 3. ; ; ; (Two segments tangent to a circle from the same exterior point are.) 4. AB = AH + HB, BC = BG + GC, CD = CF + FD, DA = DE + EA (Segment Addition) 5. AB + CD = AH + HB + CF + FD; DA + BC = DE + EA + BG + GC (Substitution) 6. AB + CD = AH + BG +GC + FD; DA + BC = FD + AH + BG + GC (Substitution) 7. AB + CD = FD + AH + BG + GC (Comm. Prop. of Add.) 8. AB + CD = DA + BC (Substitution) Statements (Reasons) 1. Quadrilateral ABCD is circumscribed about. (Given) 2. Sides,,, and are tangent to at points H, G, F, and E, respectively. (Def. of circumscribed) 3. ; ; ; (Two segments tangent to a circle from the same exterior point are.) 4. AB = AH + HB, BC = BG + GC, CD = CF + FD, DA = DE + EA (Segment Addition) 5. AB + CD = AH + HB + CF + FD; DA + BC = DE + EA + BG + GC (Substitution) 6. AB + CD = AH + BG +GC + FD; DA + BC = FD + AH + BG + GC (Substitution) 7. AB + CD = FD + AH + BG + GC (Comm. Prop. of Add.) 8. AB + CD = DA + BC (Substitution) esolutions Manual - Powered by Cognero Page 9

10 30. SATELLITES A satellite is 720 kilometers above Earth, which has a radius of 6360 kilometers. The region of Earth that is visible from the satellite is between the tangent lines and. What is BA? Round to the nearest hundredth. EB = = 7080 By Theorem 10.10,. So, is a right triangle. Use the Pythagorean Theorem. Substitute. 31. SPACE TRASH Orbital debris refers to materials from space missions that still orbit Earth. In 2007, a 1400-pound ammonia tank was discarded from a space mission. Suppose the tank has an altitude of 435 miles. What is the distance from the tank to the farthest point on the Earth s surface from which the tank is visible? Assume that the radius of Earth is 4000 miles. Round to the nearest mile, and include a diagram of this situation with your answer. Draw a circle representing the earth and choose an exterior point of the circle to represent the position of the tank. The furthest point from which the tank can be seen from the Earth s surface would be the point of tangency of a tangent segment drawn from the tank to the circle. Draw the radius to the point of tangency to create a right triangle by Theorem The length of the radius is 4000 miles and the distance to the tank is or 4435 miles km Let d be the length of one of the tangent segments from the tank to the Earth s surface. Use the Pythagorean Theorem to find d Therefore, the distance from the tank to the furthest point on the Earth s surface from which the tank is visible is about 1916 miles mi; esolutions Manual - Powered by Cognero Page 10

11 32. PROOF Write an indirect proof to show that if a line is tangent to a circle, then it is perpendicular to a radius of the circle. (Part 1 of Theorem 10.10) Given: is tangent to at T; is a radius of. Prove:. (Hint: Assume is not to.) Proof: Assume that is not to. If is not to, some other segment must be to. Also, there is a point R on as shown in the diagram such that. and are right angles by the definition of perpendicular. and. by SAS, so by CPCTC. Thus, both T and R are on. For two points of to also be on contradicts the given fact that is tangent to at T. Therefore, must be true. Proof: Assume that is not to. If is not to, some other segment must be to. Also, there is a point R on as shown in the diagram such that. and are right angles by the definition of perpendicular. and. by SAS, so by CPCTC. Thus, both T and R are on. For two points of to also be on contradicts the given fact that is tangent to at T. Therefore, must be true. 33. PROOF Write an indirect proof to show that if a line is perpendicular to the radius of a circle at its endpoint, then the line is a tangent of the circle. (Part 2 of Theorem 10.10) Given: ; is a radius of. Prove: is tangent to. (Hint: Assume is not tangent to.) Proof: Assume that is not tangent to. Since intersects at T, it must intersect the circle in another place. Call this point Q. Then ST = SQ. is isosceles, so. Since, and are right angles. This contradicts that a triangle can only have one right angle. Therefore, is tangent to. Proof: Assume that is not tangent to. Since intersects at T, it must intersect the circle in another place. Call this point Q. Then ST = SQ. is isosceles, so. Since, and are right angles. This contradicts that a triangle can only have one right angle. Therefore, is tangent to. esolutions Manual - Powered by Cognero Page 11

12 34. CCSS TOOLS Construct a line tangent to a circle through a point on the circle. Use a compass to draw. Choose a point P on the circle and draw. Then construct a segment through point P perpendicular to. Label the tangent line t. Explain and justify each step. Sample answer: 35. CHALLENGE is tangent to circles R and S. Find PQ. Explain your reasoning. Sample answer: Step 1: Draw circle A and label a point P on the circle. Step 2: Draw. (Two points determine a line.) Step 3: Construct line t perpendicular to through point P. (The tangent is perpendicular to the radius at its endpoint.) Draw perpendicular to. (Through a point not on a line exactly one perpendicular can be drawn to another line.) Since STP is a right angle, PQST is a rectangle with PT = QS or 4 and PQ = ST. Triangle RST is a right triangle with RT = PR TR or 2 and RS = PR + QS or 10. Let x = TS and use the Pythagorean Theorem to find the value of x. Sample answer: a. Draw. (Two points determine a line.) b. Construct a perpendicular at P. (The tangent is perpendicular to the radius at its endpoint.) The measure of ST is about 9.8. Since PQ = ST, then the measure of PQ is also about 9.8. Sample answer: Using the Pythagorean Theorem, x 2 = 10 2, so. Since PQST is a rectangle, PQ = x = 9.8. esolutions Manual - Powered by Cognero Page 12

13 36. WRITING IN MATH Explain and justify each step in the construction on page 720. (Step 1) A compass is used to draw circle C and a point A outside of circle C. Segment is drawn. (There is exactly one line through points A and C.) (Step 2) A line is constructed bisecting line. (Every segment has exactly one perpendicular bisector.) According to the definition of perpendicular bisector, point X is the midpoint of. (Step 3) A second circle, X, is then drawn with a radius which intersects circle C at points D and E. (Two circles can intersect at a maximum of two points.) (Step 4) and are then drawn. (Through two points, there is exactly one line.) is inscribed in a semicircle, so is a right angle. (Theorem 10.8) is tangent to at point D because it intersects the circle in exactly one point. (Definition of a tangent line.) First, a compass is used to draw circle C and a point A outside of circle C. Segment is drawn. There is exactly one line through points A and C. Next, a line is constructed bisecting. According to the definition of a perpendicular bisector, line is exactly half way between point C and point A. A second circle, X, is then drawn with a radius which intersects circle C at points D and E. Two circles can intersect at a maximum of two points. and are then drawn, and is inscribed in a semicircle. is a right angle and is tangent to. is tangent to at point D because it intersects the circle in exactly one point. 37. OPEN ENDED Draw a circumscribed triangle and an inscribed triangle. Sample answer: Circumscribed Use a compass to draw a circle. Using a straightedge, draw 3 intersecting tangent lines to the circle. The triangle formed by the 3 tangents will be a circumscribed triangle. Inscribed Use a compass to draw a circle. Choose any 3 points on the circle. Construct 3 line segments connecting the points. The triangle formed by the 3 line segments will be an inscribed triangle. Sample answer: Circumscribed Inscribed esolutions Manual - Powered by Cognero Page 13

14 38. REASONING In the figure, and are tangent to. and are tangent to. Explain how segments,, and can all be congruent if the circles have different radii. By Theorem 10.11, if two segments from the same exterior point are tangent to a circle, then they are congruent. So, and. By the transitive property,. Thus, even though and have different radii,. By Theorem 10.11, if two segments from the same exterior point are tangent to a circle, then they are congruent. So, and. Thus,. 39. WRITING IN MATH Is it possible to draw a tangent from a point that is located anywhere outside, on, or inside a circle? Explain. From a point outside the circle, two tangents can be drawn.(draw a line connecting the center of the circle and the exterior point. Draw a line through the exterior point that is tangent to the circle and draw a radius to the point of tangency creating a right triangle. On the other side of the line through the center a congruent triangle can be created. This would create a second line tangent to the circle.) From a point on the circle, one tangent can be drawn. (The tangent line is perpendicular to the radius at the point of tangency and through a point on a line only one line can be drawn perpendicular to the given line.) From a point inside the circle, no tangents can be drawn because a line would intersect the circle in two points. (Every interior point of the circle be a point on some chord of the circle.) No; sample answer: Two tangents can be drawn from a point outside a circle and one tangent can be drawn from a point on a circle. However, no tangents can be drawn from a point inside the circle because a line would intersect the circle in two points. esolutions Manual - Powered by Cognero Page 14

15 40. has a radius of 10 centimeters, and is tangent to the circle at point D. F lies both on and on segment what is the length of? A 10 cm B 16 cm C 21.8 cm D 26 cm. If ED = 24 centimeters, 41. SHORT RESPONSE A square is inscribed in a circle having a radius of 6 inches. Find the length of each side of the square. Use the Pythagorean Theorem. Apply Theorem and the Pythagorean Theorem. or about 8.5 in. So, EF = = 16. The correct choice is B. B 42. ALGEBRA Which of the following shows 25x 2 5x factored completely? F 5x(x) G 5x(5x 1) H x(x 5) J x(5x 1) So, the correct choice is G. G esolutions Manual - Powered by Cognero Page 15

16 43. SAT/ACT What is the perimeter of the triangle shown below? A 12 units B 24 units C 34.4 units D 36 units E 104 units The given triangle is an isosceles triangle, since the two sides are congruent. Since the triangle is isosceles, the base angles are congruent. We know that the sum of all interior angles of a triangle is 180. So, the measure of base angles is 60 each. So, the triangle is an equilateral triangle. Since the triangle is equilateral, all the sides are congruent. Perimeter = = 36 So, the correct choice is D. D Find each measure. 46. If an angle is inscribed in a circle, then the measure of the angle equals one half the measure of its intercepted arc. So, Substitute. 61 If an angle is inscribed in a circle, then the measure of the angle equals one half the measure of its intercepted arc. So, the arc is twice the measure of the angle. Here, is a semi-circle. So, If an angle is inscribed in a circle, then the measure of the angle equals one half the measure of its intercepted arc. So, Therefore, Since is a diameter, arc WVX is a semicircle and has a measure of 180. Use the Arc Addition Postulate to find the measure of arc VX. Therefore, the measure of arc VX is esolutions Manual - Powered by Cognero Page 16

17 In, GK = 14 and. Find each measure. Round to the nearest hundredth. 50. METEOROLOGY The altitude of the base of a cloud formation is called the ceiling. To find the ceiling one night, a meteorologist directed a spotlight vertically at the clouds. Using a theodolite, an optical instrument with a rotatable telescope, placed 83 meters from the spotlight and 1.5 meters above the ground, he found the angle of elevation to be. How high was the ceiling? 47. If a diameter (or radius) of a circle is perpendicular to a chord, then it bisects the chord and its arc. So, bisects Therefore, 71 First, use a trigonometric ratio to find the value of x in the diagram. 48. JK If a diameter (or radius) of a circle is perpendicular to a chord, then it bisects the chord and its arc. So, bisects 7 Then JK = 7 units. 49. Here, segment HM is a diameter of the circle. We know that. The height of ceiling is about or meters. 109 about m esolutions Manual - Powered by Cognero Page 17

18 Determine whether the triangles are similar. If so, write a similarity statement. Explain your reasoning. 53. Solve each equation. 51. Yes; From the markings,. So, EAB and DBC are congruent corresponding angles. By the reflexive property, C is congruent to C. Therefore, by AA Similarity. Yes; by AA Similarity Yes; Since the ratio of the corresponding sides,,, and all equal, the corresponding sides are proportional. Therefore, by SSS Similarity. Yes; by SSS Similarity esolutions Manual - Powered by Cognero Page 18

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